FREQUENCY-DEPENDENCE OF SHAPIRO STEPS IN JOSEPHSON-JUNCTION ARRAYS

We present an analytic model describing the dynamics of an N x N square array of overdamped Josephson junctions in a magnetic field producing 1/2 flux quanta per unit cell (f = 1/2) biased with combined direct and alternating currents injected parallel to a row of junctions. In agreement with recent experiments and numerical simulations, we find Shapiro steps at voltages corresponding to integer multiples (giant steps) and half-integer mutliples (fractional giant steps) of N times the corresponding single-junction Shapiro-step voltage. In the low-drive-frequency limit, our calculations show that each 2 x 2 group of cells in the array can be approximated by a single junction, leading to integer and half-integer Shapiro steps. At higher drive frequencies, a more complicated behavior of the 2 x 2 cell leads to a suppression of half-integer steps relative to integer steps. In a high-frequency expansion, we find the maximum width of a half-integer step to be inversely proportional to the ac drive frequency.